Two-scale homogeneous functions in wavelet analysis

Two-scale homogeneous functions are functions h satisfying h(x) = λh(2x) for some constant λ. They form a class of special functions and include homogeneous polynomials. In this article we investigate two-scale homogeneous functions that are contained in a shift-invariant space S(φ) where φ is an r-vector of functions and satisfies a vector refinement equation. The structure of these functions is analyzed. In particular, we establish a one-to-one correspondence between these two-scale homogeneous functions with order λ and the left eigenvectors of a finite matrix (derived from the mask for φ) associated with eigenvalue λ. As a corollary, we show that if φ is supported in [0, N], and provides accuracy m (i. e., {1, x,..., x^m-1} ⊂ S(φ)), then 1, 1/2,..., 1/2^m-1 are eigenvalues of an rN × rN matrix. This is used to prove that among all the refinable vectors of functions with a fixed support, the B-spline vector with uniform multiple knots yields the optimal accuracy. Thus a conjecture of Plonka is confirmed. The main difficulty we overcome here is that the support of the mask may be larger than that of φ in the vector case. (The mask can even be infinitely supported). Another application is to show that subdivision operators (related to the adjoint of the Ruelle operators) do not have eigenvalues in ℓ_p (ℤ). This improves the spectral analysis of subdivision operators. The reconstruction of φ from two-scale homogeneous functions is also considered in the scalar case. This reconstruction is possible globally only if φ is a B-spline. But if we want to reconstruct the pieces of φ on integer intervals, then we can always do so. Our study leads to further more questions concerning the relation between two-scale homogeneous functions and refinable functions.